Simplify and expand the following expression: $ \dfrac{2x}{4x - 10}+\dfrac{3x - 7}{3x - 1} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4x - 10)(3x - 1)$ Multiply the first term by $\dfrac{3x - 1}{3x - 1}$ $ \begin{align*} \dfrac{2x}{4x - 10} \times \dfrac{3x - 1}{3x - 1} & = \dfrac{(2x)(3x - 1)}{(4x - 10)(3x - 1)} \\ & = \dfrac{6x^2 - 2x}{(4x - 10)(3x - 1)}\end{align*} $ Multiply the second term by $\dfrac{4x - 10}{4x - 10}$ $ \begin{align*} \dfrac{3x - 7}{3x - 1} \times \dfrac{4x - 10}{4x - 10} & = \dfrac{(3x - 7)(4x - 10)}{(3x - 1)(4x - 10)} \\ & = \dfrac{12x^2 - 58x + 70}{(3x - 1)(4x - 10)}\end{align*} $ Now we have: $ = \dfrac{6x^2 - 2x}{(4x - 10)(3x - 1)} + \dfrac{12x^2 - 58x + 70}{(3x - 1)(4x - 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{6x^2 - 2x + 12x^2 - 58x + 70}{(4x - 10)(3x - 1)} $ $ = \dfrac{18x^2 - 60x + 70}{(4x - 10)(3x - 1)}$ Expand the denominator: $ = \dfrac{18x^2 - 60x + 70}{12x^2 - 34x + 10}$ Simplify: $ = \dfrac{9x^2 - 30x + 35}{6x^2 - 17x + 5}$